9.3 Consensus for Individuals with No Stubbornness
9.3.1 Stability and Convergence to Consensus
The convergence result on the fundamental algorithm Eq. (9.2), where all individuals are maximally open to influence, i.e. not stubborn: bi =0,∀i∈ I, is first presented. The key result is summarised in the following theorem.
Theorem 9.1. Let C, which satisfies Assumption 9.1, and G = (V,E,A)be given. Then for all initial conditions x(0), the social network, with each individuals’ opinions evolving according to Eq. (9.2), reaches a consensus on all topics exponentially fast5 if and only if
Re((1−λi(L))λk(C))<1, ∀i=2, ...,nandk∈ J, (9.11)
where λi(L) the eigenvalues of the Laplacian matrix L, with λ1(L) = 0. Moreover, the
solution satisfies
lim
t→∞xi(t) = (γ⊗ξ)
>
x(0)ζ,∀i∈ I, (9.12) where γ> ∈ R1×n is the normalised left eigenvector of L associated with the 0 eigenvalue satisfyingγ>1n=1, andξ>andζ are eigenvectors ofC defined in Assumption 9.1.
Proof. First, observe that Eq. (9.11) holds only ifλi(L),i=2, ...,nare nonzero, which in turn holds if and only if the graphGhas a directed spanning tree (see Lemma A.8). That is, satisfying Eq. (9.11) implies that G has a directed spanning tree6. We first establish the sufficiency of Eq. (9.11).
With all individuals maximally open to influence, i.e. bi = 0∀i∈ I, the opinions
x(t) of the social network evolve according to Eq. (9.5). Denote M = −Ind+ (In−
L)⊗C. Clearly theith eigenvalue of M is equal to −1+λ
i(A) where λi(A) is the
ith eigenvalue of A= (In− L)⊗C. The associated eigenvector isvi, wherevi is the eigenvector ofAassociated withλi(A). One has thatλi(A) =µkϕl, whereµk andϕl are eigenvalues of In− L andC respectively, k ∈ I, l ∈ J (see Lemma A.5). Then, one can verify thatvi = uk⊗wl is an eigenvector of Aassociated withλi(A), where
uk and wl are eigenvectors of In− L andC associated with µk and ϕl, respectively. According to Assumption 9.1, C has a single eigenvalue at 1, which is denoted as ϕ1. IfG has a directed spanning tree, then Lhas a single eigenvalue at zero, which
implies In− L has a single eigenvalue at 1. Denote this eigenvalue as µ1. Then
clearly, λ1 = µ1ϕ1 = 1 is an eigenvalue of A with right eigenvector v1 = 1n⊗ζ.
5Convergence is of course exponential, since Eq. (9.5) is an LTI system.
6The requirement of a directed spanning tree is not surprising, and appears frequently in consensus
problems for multi-agent systems [Ren and Cao, 2011]. IfGdoes not have a directed spanning tree then
there is at least one individuali, or a group of individuals forming a closed and strongly connected
subgraph (see Appendix A.2), who does not consider any other opinions. It follows that for generic ini-
tial conditions, individuali’s opinions (or the closed group of individuals’ opinions) evolve separately,
For λi = µ1ϕl,l = 2, . . . ,d, clearly λi = ϕl has real part strictly less than 1, because Assumption 9.1 has that Re(ϕl) < 1. For λi = µkϕl where k ∈ {2, . . . ,n},l ∈ J, if Eq. (9.11) is satisfied then λi has real part strictly less than 1. This in turn implies that Re(λi) < 1 for all i 6= 1, which implies that all eigenvalues of M have strictly negative real part, except for a single eigenvalue at the origin, with associated right eigenvector v1 =1n⊗ζ. From linear systems theory, one has that x(t) = eMtx(0) = PeJtP−1x(0), wherePis such thatJ =P−1MP, and J is the Jordan canonical form of M, ordered such that
J = " 0 0>(nd−1) 0(nd−1) ∆ # . (9.13)
The nd−1 nonzero diagonal entries of∆ are the stable eigenvalues of M. One then obtains limt→∞x(t) = p1q1>x(0) where p1 and q>1 are right and left eigenvectors
of M associated with the single zero eigenvalue, satisfying p>1q1 = 1. The above analysis yielded p1 = 1n⊗ζ. One can easily verify that q>1 = (γ⊗ξ)> and thus
limt→∞x = (γ⊗ξ)>x(0)(1n⊗ζ). In other words, limt→∞xi(t) = xj(t) = αζ as in
Eq. (9.12), with α= (γ⊗ξ)>x(0). Eq. (9.6) is satisfied. The sufficiency of Eq. (9.11)
has thus been established.
It remains for the necessity of Condition 9.11 to be established. Suppose that Eq. (9.11) is not satisfied. Then there is some λi = µkϕl,k ∈ {2, . . . ,n},l ∈ J such that the eigenvalue of M, −1+λi, is in the closed right half-plane. The system is either unstable, or −1+λi is on the imaginary axis (possibly at the origin). In the latter case either a) there are now at least 2 eigenvalues of M at the origin, or b)
M has a pair of purely imaginary eigenvalues. Regarding a), the system is either unstable (there is a Jordan block in J associated with the eigenvalue 0, of size at least 2×2), or there are at least two 1×1 Jordan blocks associated with a zero eigenvalue. In the second possibility, consider for convenience the case where there are precisely 2 zero eigenvalues. Then, x converges exponentially fast to a subspace spanned by
{v1,vi}where vi is an eigenvector of M associated with eigenvalue λi = µkϕl = 0,
k 6= 1. Becausek 6= 1,vi =uk⊗wl cannot take the form1n⊗wl, for somewl ∈ Rd, which implies that consensus is not reached for generic initial conditions. Regarding
b), denote one of the imaginary eigenvalues asλi = µkϕl, and recall thatk 6=1. Then, the system oscillates but not in consensus because, similar to the above arguments, vi associated with the imaginary λi cannot take the form 1n⊗wl. The proof is complete.
Remark 9.3. Theorem 9.1 provides a necessary and sufficient condition for the stability of the opinion dynamics system. Clearly, whether consensus is achieved depends on the combination of the network topology, as encoded byL, and the logical interdependence as encoded by C. Given a C satisfying Assumption 9.1 , two different graphs G1 and G2 may have different
stability properties. For networks with no stubborn individuals, this is in direct contrast to the discrete-time result in [Parsegov et al., 2017], which establishes that consensus is reached if and only if C is regular, and either limk→∞Ck = 0n×n orW is fully regular7.
7A matrixAis regular if lim
Here,W is the influence matrix, i.e. the discrete-time counterpart to L. Thus, in discrete- time, consensus is guaranteed if both C andW separately satisfy the required conditions, whereas in the continuous-time model Eq. (9.4), the conditions intertwineC andL, making prediction of conditions for consensus significantly more difficult. Naïve analysis of Land I−C separately could lead to a conclusion that a consensus is reached, when in fact the system is unstable; this is illustrated in the simulations in Section 9.5.
The author conjectures that the difference between the continuous- and discrete-time mo- dels is due to the different assumptions on the logic matrix and network topology. In discrete- time, W is row-stochastic (i.e. kWk∞ = 1) and C is regular (which implies kCk0 ≤ 1 for some matrix norm k · k0). In continuous-time, there are no restrictions on the size of the
entries in A(and consequently no restrictions on the size of the entries in L), and C only needs to satisfy Assumption 9.1. This means separately, the cognitive process Eq. (9.8) and the consensus process (which involvesL) can have much larger oscillations (i.e. the complex eigenvalues have large moduli) in continuous-time. When the two processes occur separately, there is no risk of instability, but when individuals discuss interdependent topics simultane- ously, the two processes may combine to cause instability. Remark 9.5 below explores this in more detail.
It may be difficult to verify the conditions in Theorem 9.1 for complex networks because the sprecise values of eigenvalues of both L,C are needed. Two results are now presented on sufficient conditions which guarantee consensususing limited information about the network and the logic structure. The detailed motivation for this is discussed in Remark 9.5 below.
Corollary 9.1. LetG = (V,E,A)be given and suppose thatG has a directed spanning tree. Then for any given C satisfying Assumption 9.1, there exists a graph with the same node and edge set asG but with different edge weights, G = {V,E,A}, such that consensus of opinions is achieved using Eq. (9.2).
Proof. LetLbe the Laplacian associated with the graphG. One can easily verify that Re((1−λi(L))λk(C)) =dk−yidk±ziek, (9.14)
where, without loss of generality, λi(L) = yi ±zi and λk(C) = dk ±ek are com- plex conjugate eigenvalues of Land C respectively. Here, zi,ek > 0, andyi > 0 for all i ≥ 2 since G has a directed spanning tree (see Lemma A.8). Because Assump- tion 9.1 holds, dk ≤ 1, for all k. For i = 2, . . . ,n and k = 1, . . . ,d, it follows that Re((1−λi(L))λk(C)) < 1 ⇔ dk−yidk+ziek < 1. Define A = αA, where α > 0 is a constant which adjusts every edge weight. Let L be the Laplacian associated with G = (V,E, ¯A). Observe thatRe (1−λi(L))λk(C) = Re((1−αλi(L))λk(C)). It follows that consensus of opinions is achieved on the graph G if and only if
dk−α(yidk−ziek) < 1. Since dk ≤ 1 under Assumption 9.1, there always exists a sufficiently small α for which dk−α(yidk−ziek) < 1 holds, because yi > 0∀i ≥ 2. The proof is complete.
The above corollary states that for any given C, there always exists a graph G
for which consensus of opinions can be achieved. The proof gives a simple way to scale the edge weights, as captured by A, by the same constant α. It is however, not the only way to scale the edge weights. Below, a more complex scaling method is presented, which requires only limited knowledge of the network topology and the logic structure C. The reader is encouraged to revisit the Geršgorin disc theorem, viz. Theorem A.3, as it will be used here and later in several other instances in this chapter.
Corollary 9.2. LetC, which satisfies Assumption 9.1, andG = (V,E,A)be given. Suppose thatGhas a directed spanning tree. Then consensus of opinions is achieved if8, for all k∈ J
¯ l<min |1− |λk|cos(θk)|(1+cos(θk)) |λk|sin2(θk) , 0.5 , (9.15)
where|λk| = |λk(C)|andtan(θk) = ek/dk with λk(C) = dk±ek. Here, l¯= maxi∈Ilii,
and lii =∑j∈Niaij is the i
th diagonal entry ofL.
Proof. The system Eq. (9.5) reaches a consensus if and only if Statement 9.11 in The-
orem 9.1 is satisfied. This is equivalent to ensuring that
dk−yidk+ziek <1 (9.16) whereλi(L) =yi±ziandλk(C) =dk±ekare any eigenvalue ofLandC, respecti- vely, except for λ1(L) =0 andλ1(C) =1. According to Assumption 9.1, dk <1.
The definition of Limplies that it has nonnegative diagonal entries and nonpo- sitive off-diagonal entries, and moreover each row sums to 0 (see Appendix A.2). Moreover, Lhas precisely one eigenvalue at 0 sinceG contains a directed spanning tree (see Lemma A.8). Combining these observations with Theorem A.3, one con- cludes that every nonzero eigenvalue of Lis contained in the disc centred at ¯l, with radius ¯l. Denote this disc as Dl¯. The fact that ¯l < 0.5 implies yi < 1 (from Theo- rem A.3). Thus, dk−yidk < 1 because dk < 1 . If λk(C) is real, i.e. ek = 0, then Eq. (9.16) is satisfied. If all eigenvalues ofC are real, then ¯l<0.5 ensures stability.
Consider now ek > 0 for some k. Observe that Eq. (9.16) is implied by zi2ek2 < (1−dk+yidk)2, which is in turn implied by
¯
z2iek2 <(1−dk+yidk)2, (9.17) where ¯zi ≥ zi is such that βi = yi+z¯i is on the boundary of Dl¯. Because βi is on the boundary of D¯l, it satisfies (yi−l¯)2+z¯2i = l¯2, which yields ¯z2i = −yi2+2yil¯. Substituting into Eq. (9.17) yields
(−yi2+2yil¯)ek2 <(1−dk+yidk)2. (9.18)
8As will be evident in the corollary proof, the right of Eq. (9.15) is well defined, since the left term
approaches(1+|λk|)/2|λk|>1/2 asθk→π. This indicates that Eq. (9.15) holds for real eigenvalues ofC, i.e.θk={0,π}.
Expanding and rearranging for ¯lyields ¯l< 21y i (1−dk)2 ek2 + dk(1−dk) ek2 + yi 2 (dk2+ek2) ek2 , or ¯ l< fk(yi) |λk|2sin2(θk) , (9.19) where fk(yi) = (1−ac)2+yi2c2+2yiac(1−ac) 2yi , (9.20)
with a = cos(θk) and c = |λk|. Recall that yi > 0. Calculations show that ¯yi =
|1−ac|/c > 0 is a unique minimum of fk(yi)for yi ∈ (0,∞). Since the right hand side of Eq. (9.20) is always strictly positive, it follows that Eq. (9.19) is implied by ¯
l< fk(y¯i)/|λk|2sin2(θk), which after some rearranging yields
¯
l< |1− |λk|cos(θk)|(1+cos(θk)) |λk|sin2(θk)
. (9.21)
The proof is completed by noting that Eq. (9.21) must hold for allkto guarantee that Eq. (9.15) holds. Note that|1− |λk|cos(θk)| 6=0 because|λk|cos(θk) =dk <1.
Consider the scenario whereC(η)varies smoothly as a function of some parame- terη∈ [a,b], and for someκ∈ (a,b), λp(C(κ))has negative real part. Suppose furt- her that λp(C(η))is real for η ≤ κ, and is complex for η >κ. Then, limη→κθp = π. Notice that, separately, limθp→π1+cos(θk) = 0 and limθp→πsin2(θk) = 0. As will now be shown, Eq. (9.15) continues to hold, i.e. is evaluable, as θp approaches π. Define g(θp) = |λp|sin2(θp) and h(θp) = (1− |λp|cos(θp))(1+cos(θp)). Denote limθp → π− as the limit of θp approaching π from the left. Since h(θp),g(θp) are continuous inθp, calculations using L’Hôpital’s rule yield
lim θp→π− h(θp) g(θp) = θp→limπ− h0(θp) g0(θp) = 1+|λp| 2|λp| . (9.22)
That is, the limit exists. This is consistent with Eq. (9.15) because(1+|λp|)/2|λp|> 1/2 for|λp|>0.
Remark 9.4. Corollary 9.2 is a stronger result than Corollary 9.1. Corollary 9.1 requires the scaling of every aij by the same constantα>0. Corollary 9.2 shows that aij only need to be
adjusted for individual i if lii =∑nj∈Niaij exceeds the right hand side of Eq. (9.15). Moreover,
for individual i, aij do not need to be scaled by the same constant for different j.
Remark 9.5. Checking the sufficiency condition in Corollary 9.2 requires only limited in- formation concerning LandC. Furthermore, consider the first term on the right hand side of Eq. (9.15). One can rewrite this as (1−dk)(1+cos(θk))
eksin(θk) . Sets of topics whose C have large
ek andθk close to π/2 are associated with a cognitive process, described in Eq. (9.8), where
the opinions for an isolated individual would oscillate heavily and rapidly before settling to a consistent belief system. Given such aC, Corollaries 9.1 and 9.2 show it is always possible to reach a consensus if there is a sufficiently slow exchange of opinions (weights aij are small).
The cost of guaranteeing consensus is a slower speed of convergence. On the other hand, rapid discussions (arising from large aij values) can create large oscillations in the
opinions. When topics are uncoupled, i.e. C= Id, there is no risk of instability with large aij,
provided the graph has a directed spanning tree. However, when the topics are coupled, large oscillations in both the cognitive process and in the consensus process may lead to a collapse in the discussions. This is the marriage of two aspects of the opinion dynamical system, viz. network topology and logical interdependence, as alluded to in Sections 9.1 and 9.2. This is of particular interest becauseCrelates to a cognitive process and thereforecannot be easily changed for a given set of issues. It also provides commentary on the relative time scale (or intensity) of the cognitive process, i.e. Eq. (9.8), and the interpersonal interactions arising from the network, when establishing conditions required for consensus. Corollary 9.2 provides a straightforward method to determine the class ofLwhich ensures consensus of opinions for a givenC.